Chuck has 20 feet of fencing and wishes to make a rectangular fence for his dog Rover. If he uses his house for one side of the fence what is maximum area?

50 square feet.

Step-by-step explanation:

Let x represent the width of the fence and y represent the length of the fence.

One side of the length is already covered by the house. This leaves 2x and y. This must sum to 20. Hence:

The area of the enclosure will be given by:

From the first equation, we can subtract 2x from both sides:

Substitute:

This is now a quadratic. Recall that the maximum value of a quadratic always occurs at its vertex. So, find the vertex of the equation. Distribute:

In this case, a = -2, b = 20, and c = 0.

Hence, the vertex is:

Substitute this value back into the equation and evaluate:

The maximum area is 50 square feet.

This occurs when the dimensions are 5 feet by 10 feet.

thus, the first answer choice, y = sec x, is the correct one